Teichmüller space for hyperkähler and symplectic structures

Let SS be an infinite-dimensional manifold of all symplectic, or hyperkähler, structures on a compact manifold MM, and Diff0 the connected component of its diffeomorphism group. The quotient S/Diff0 is called the Teichmüller space of symplectic (or hyperkähler) structures on MM. MBM classes on a hyperkähler manifold MM are cohomology classes which can be represented by a minimal rational curve on a deformation of MM. We determine the Teichmüller space of hyperkähler structures on a hyperkähler manifold, identifying any of its connected components with an open subset of the Grassmannian variety SO(b2−3,3)/SO(3)×SO(b2−3)SO(b2−3,3)/SO(3)×SO(b2−3) consisting of all Beauville–Bogomolov positive 3-planes in H2(M,R)H2(M,R) which are not orthogonal to any of the MBM classes. This is used to determine the Teichmüller space of symplectic structures of Kähler type on a hyperkähler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes v∈H2(M,R)v∈H2(M,R) with q(v,v)>0q(v,v)>0, where qq is the Bogomolov–Beauville–Fujiki form on H2(M,R)H2(M,R).